General Robert E. Lee
and Modern Decision Theory

One of the classic campaigns in the annals of military history was waged at
Chancellorsville, Virginia, in May 1863 between the Army of the Potomac, led by
Major General Joseph L. Hooker, and the Army of Northern Virginia, commanded by
General Robert E. Lee. During the campaign, Lee, with a force approximately
half the size of Hooker’s, repulsed the North’s advance into Virginia and
achieved a strategic victory that has been studied by students of military art
throughout the world. However, today’s critics of the quantitative-oriented
decision tools being used by our military services say that this battle would
never have transpired if these same tools had been used then.1 They
feel that under the present decision-making process Lee would not have met
Hooker’s advance but instead would have retreated to southern Virginia or even
into North Carolina. Contrary to that course, Lee decided to give battle, and
he won a brilliant victory.

The question to which we must address ourselves, then, is this: Was Lee’s
decision to fight based strictly on native intuition—leaving quantitative
analysis nothing to offer—or could it be rationally justified by using modern
decision techniques? This article argues that there are decision tools in-being
today that can be used to support Lee’s decision. Whether Lee applied such
tools, either consciously or subconsciously, is not known, but we do know that
he was no stranger to the science of numbers. Douglas Southall Freeman, who
spent over twenty years studying the life of the great Confederate commander,
declared: “His mind was mathematical and his imagination that of an engineer.”2

Lee’s background supplies ample evidence to confirm this evaluation. He
graduated second in the class of 1829 from West Point, which was at that time
primarily an engineering school. So proficient had he been in the field of
mathematics that he was appointed acting assistant professor to instruct other
cadets when he was only a second-year student. After graduation he entered the
Corps of Engineers, and subsequent years found him working on engineering
projects throughout the United States. It is doubtful if a person as familiar
with numbers as Lee would not either explicitly or implicitly have quantified
at least partially the alternatives open to him at Chancellorsville.

In the following sections the reader will find descriptions of three
decision tools that could have been applied by Lee to support his decision to
fight at Chancellorsville. These tools are the Lanchester equations, Bayes’
theorem, and the von Neumann-Morgenstern utility theorem. Before these decision
tools are outlined, however, a brief description of the battle of
Chancellorsville may prove useful.

The Battle of Chancellorsville

Probably the most comprehensive and unbiased study of this battle appears in
The West Point Atlas of American Wars, edited by Colonel Vincent J. Esposito,
former Professor of Military Art and Engineering at West Point.3 The
following description draws heavily upon that fine work.

In April 1863 the newly appointed commander of the Army of the Potomac,
General Hooker, with 118,000 men, faced General Lee’s Army of Northern
Virginia, approximately 60,000 strong, across the Rappahannock River at
Fredericksburg, Virginia. On the 29th and 30th Hooker moved approximately
73,000 troops on a wide flanking movement across the Rappahannock to the
vicinity of Chancellorsville to attack Lee from the rear. To hold Lee in
position, Major General John Sedgwick, U.S. Army, with the remaining 45,000,
maintained his position opposite Fredericksburg. (Figure 1a) Although Hooker’s units were
in position on the 30th, he awaited further reinforcements and did not advance
from the vicinity of Chancellorsville until the first of May.

By this time Lee had interpreted Hooker’s strategy. Leaving Major General
Jubal Early, C.S.A., with 10,000 men to face Sedgwick, Lee moved his units
toward Chancellorsville. The first clash occurred the afternoon of the first,
and Hooker, apparently having lost his courage, gave up the initiative and
recalled his much larger force to Chancellorsville into a defensive position.

That night Lee and Lieutenant General “Stonewall” Jackson, aware of Hooker’s hesitancy, conceived a daring plan.
Lee would maintain his position with approximately 17,000 men and demonstrate
against Hooker’s front, while Jackson would take the remaining force, using
Major General Jeb Stuart’s cavalry as a screen, and turn the enemy flank.
(Figure 1b)

The movement took the better part of the next day, but shortly before sundown
Jackson struck Hooker’s exposed flank. The battle raged during the night until
the Federal Army gave way before Jackson’s thrusts. The sensation of victory
that Lee felt, however, must have been more than overshadowed by the loss of
Jackson, who had ridden too far forward in reconnoitering the Union positions
and had been shot by mistake when returning to his own lines.

On the third of May, Hooker again failed to take the initiative against Lee’s
split army, and although he was wounded later in the day by cannon fire, he
would not relinquish command to his subordinate. By sundown Lee had united his
separated units and was pushing Hooker back against the Rappahannock. But Lee’s
troubles were not over. Earlier that day Sedgwick had attacked at Fredericksburg,
overrun Early’s weak position, and was marching toward Lee’s rear.

Figure 1 The battle of Chancelorville

Again counting on Hooker’s hesitancy, Lee reversed his field, leaving Jeb
Stuart with 25,000 men to face Hooker’s 73,000, and marched the remaining units
toward Sedgwick’s advancing army. Another flanking movement, using Early’s
remaining force, proved successful, and the morning f the fifth found Sedgwick
back across the Rappahannock. (Figure 1c)

Lee, determined to crush Hooker, again reversed his field. But Hooker had
had enough. On the sixth of May he withdrew his forces across the river before
Lee could accomplish this objective.

The Lanchester Equations

A rather mathematical approach to the problem of battle decisions was
provided by Frederick Lanchester in his article, “Mathematics in Warfare.”4
He derived two basic equations relating numerical strength and another
constant, which he called “fighting value,” to total strength. These equations
can be adapted to the present analysis if we let “fighting value” represent the
aggregate of all factors affecting the battle other than numerical strength.

Lanchester assumed that the number of men killed or incapacitated per unit
time during a battle is directly proportional to the strength of the opposing
force. This can be shown mathematically as

in which b and r
represent the numerical strengths of the Blue and Red forces, respectively; t
is time; and Kb and Kr are the fighting values of the two
units.

If Kb= Kr, the battle depends entirely on the numerical
strengths of the two forces. If Blue has twice as many men as Red, the ensuing
battle is as depicted in Figure 2a. When Red’s force has been completely
annihilated, Blue will have 866 men remaining.

Incidentally, this also shows the value of concentration. If Red originally
had 1000 men and separated them into two armies, and each gave battle in turn,
Blue would have 866 men after destroying the first Red army—enough easily to
defeat Red’s second force, all other things remaining equal.

If the values Kb and Kr are not equal, then these,
too, must be considered in equating the total strengths of the forces. For the
condition of equality, losses must be proportional to numerical strengths:

In words, the total strengths of the two forces are equal when the squares
of the numerical strengths, multiplied by the fighting values of the units, are
equal. This is what Lanchester called the “n-square Law.”

The effect of concentration versus separation of forces has already been
mentioned. Lanchester also gave a mathematical relationship for the aggregate
numerical strength of the separated forces. (Figure 2b) Let the numerical
values of the Blue and Red forces be represented by lines b and r. In an
infinitesimal interval of time the change in b and r will be represented
by db and dr in the relationship:

Since in the “n-square Law” we are interested in the squares of the
strengths, we here note what happens to the change of the area of b2
and r2 when the increments db and dr are subtracted.
The change in b2 is 2bdb and the change in r2 is 2rdr.
According to equation (5) these are equal, so the difference between the two
squares is constant.

r. represents numerically a second Red army of the strength necessary
in a separate action to place the Red forces on equal terms with the Blue
force. Graphically, Red’s total numerical strength is the hypotenuse of a right
triangle, the legs of which are the two separate forces. (Figure 2c)

Now if Lee had had available Lanchester’s equations (4) and (6), he could
have mathematically verified his decision to fight. First, let us compare the K
values. The battle of Fredericksburg, which was the last engagement between the
two armies, provides a starting point. The numerical strengths and losses of
the Northern forces were both twice that of the South. Accordingly, equation
(3) is satisfied, and there existed an equality in the total fighting strength
of both sides. By equation (4),

Lee had a four-to-one advantage in “fighting value.”

At the time of Lee’s critical decision, Hooker had divided his army into two
forces. One force of approximately 45,000 men under Sedgwick was left to
contain Lee, while Hooker, with 73,000 men, effected a flanking maneuver to
attack Lee’s rear. In the meantime, however, Lee had left 10,000 men in place
under Early to face Sedgwick and took 50,000 men to meet Hooker’s main thrust.
According to equation (6), the proportional numerical strengths were then:

Lee’s total strength was greater than Hooker’s!

I hesitate to push this approach too far. The K values were derived from
only one campaign and would require further verification. The analysis is
predicated on the assumptions that the separated forces give battle in turn and
that combat takes place in the open. The first assumption was fulfilled at
Chancellorsville, but the second might prove difficult to verify. The approach
does show, however, that the Lanchester equations, even if indiscriminately
applied, could be used to support Lee’s decision.

Bayes’ Theorem

Bayes’ theorem can be utilized to refine any hypothesis that Lee might have
held about defeating the Northern forces. One version of this theorem takes the
form:

If Lee had placed a certain a priori probability on the hypothesis that he
could defeat the Northern army, and if the probability of winning a battle,
given that the hypothesis was true, was relatively high whereas the normal
probability of winning was relatively low, then given the past event—the battle
of Fredericksburg (or better yet, eleven wins in thirteen encounters)—his a
posteriori probability of the hypothesis would be greater and more
meaningful than his a priori probability.

For example, let us say Lee placed a .3 probability on the hypothesis that
he could defeat the Union force. If the probability of winning at
Fredericksburg, given the hypothesis was true, was .6, and a normal probability
of winning at Fredericksburg was .2, then

His a priori probability of winning was .3, but with the use of additional
information (past events), this probability increased to .9. He would now have
greater faith in his original hypothesis that he could defeat the Union army
and might therefore decide to meet Hooker’s advance.

The Von Neumann-Morgenstern
Utility Theorem

Professors John von Neumann and Oskar Morgenstern have shown that under
certain circumstances it is possible to construct a set of numbers for a
particular individual that can be used to predict his choice in uncertain
conditions. Briefly, this theorem states that if an individual can rank three
commodities in an order of preference, say A>B>C, then in a choice
between a certain prospect containing B and an uncertain prospect containing
A
and C with a probability, p, of getting A, there is a value p
which makes the individual indifferent between the two prospects. Two of the
commodities can be given arbitrary values, and once the individual provides the
probability, p, which makes him indifferent between the two prospects,
the value of the third commodity can be obtained. These values will then have
certain cardinal properties that can be used to evaluate the decision process.

Napoleon stated that “the General is the head, the whole of the Army.”5
If Napoleon’s maxim is correct, Lee’s decision to fight could have been
predicated on a comparison of the high-level commanders of the two armies. He
did know a majority of the commanders on both sides. Of the eight corps
commanders under Hooker, five had served with Lee in the Mexican War and two
had been cadets at West Point when Lee served as superintendent of that
institution from 1852 to 1855. Aligned against these commanders, Lee had the
following men who would playa significant role in the coming battle: Jackson,
the trusty lieutenant who had more than proved himself in previous campaigns;
Stuart, the dashing cavalry officer who had highly impressed Lee as a cadet at
West Point; and Early, an 1837 classmate of Hooker and a veteran of the Mexican
War.

That Lee had definite opinions about the abilities of his enemy is apparent
from the letters of that day. Previously, when McClellan had been replaced as
commander of the Army of the Potomac, Lee expressed sorrow that his old
associate of the Mexican War would no longer oppose him: “We had always
understood each other so well. I fear they may continue to make these changes
till they find someone whom I don’t understand.”6 When Hooker
replaced Burnside as commander of the federal forces, Lee accepted the change
with complacency. In his personal letters, however, he jested mildly over the
apparent inability of Hooker to determine a course of action.7

Contrasted with this rather low opinion of the opposition leader, we find
this lofty estimate of Jackson’s capabilities: “Such an executive officer the
sun never shone on. I have but to show him my design, and I know that if it can
be done, it will be done. No need for me to send or watch him. Straight as a
needle to the pole he advances to the execution of my purpose.”8

This intimate knowledge of the opposing commanders and definite opinion of
their capabilities belong in Lee’s calculus. Given this, he could have used the
von Neumann-Morgenstern utility theorem to establish a quantitative comparison
of the leadership abilities of both sides. As an example, let us say that Lee
would rate the three commanders, Hooker, Jackson, and Stuart, in the following
order: Jackson>Stuart>Hooker. We now set any arbitrary value for Jackson,
say 100, and Stuart, say 90, and then determine at what probability, p,
Lee would be indifferent between the certain prospect of getting Stuart and the
uncertain prospect which, if selected, provided the probability, p, of
getting Jackson.

Numerical values of the capabilities of the other commanders could be
derived in the same manner. These values could then be aggregated to give a
rough quantitative comparison of Lee’s view of the opposing leadership
abilities. This comparison would provide an important input to the
decision-making process.

This article describes how three modern quantitative tools could have been
employed by General Robert E. Lee to aid in the critical decision facing him on
the eve of the battle of Chancellorsville. This survey of decision tools is
certainly not exhaustive—there are others that one could utilize. There are also
other inputs that belong in Lee’s calculus, such as the “super” image of Lee
that had been created, the effect of the recently issued Emancipation
Proclamation in hardening Southern resistance, and the comparative morale in
the two armies.

The point to be emphasized, however, is that any tool, quantitative or
otherwise, which aids the decision-maker in his choice, not only should but
must be employed. If that choice is among a number of alternatives, however,
systematic quantitative analysis will prove essential in delineating clearly
the basic relationships and interactions between the many diverse factors that
the decision-maker must consider. It will prove even more essential in the
military than in the business world, where the forces of competition working
through the price mechanism furnish a reliable guide to planning. In matters of
national security no such mechanism is available.

This does not mean that sound judgment has been replaced by the computer. As
far as I can determine, no one has ever advocated the exclusive use of
mathematical tools in the determination of policy. Surely this was not the
theme of the Hitch and McKean book, which had such a powerful impact upon
defense strategy:

Economic choice is a way of looking at problems and does not necessarily
depend upon the use of any analytical aids or computational devices. . . .
Where mathematical models and computations are useful, they are in no sense
alternatives to or rivals of good intuitive judgment; they supplement and
complement it. Judgment is always of critical importance in designing the
analysis, choosing the alternatives to be compared, and selecting the
criterion. Except where there is a completely satisfactory one-dimensional
measurable objective (a rare circumstance), judgment must supplement the
quantitative analysis before a choice can be recommended.9

The responsibility of decision still rests with the commander.
Quantitative analysis does not relieve him of that responsibility, but it can
make that responsibility less formidable. Hq Pacific Air Forces

Notes

1. One such view was expressed by Colonel Francis X. Kane in “Security Is
Too Important to Be Left to Computers,” Fortune, April 1964.

Lieutenant Colonel Herman L. Gilster (USMA; Ph.D., Harvard
University) is assigned to the Directorate of Operations Analysis, Hq PACAF.
Except for a year as Chief, Tactical Analysis Division, Hq Seventh Air Force,
PACAF, he was on the faculty of USAF Academy, 1963-71, as Associate Professor
of Economics and Management. His articles have been published in Papers in
Quantitative Economics (University of Kansas Press), Air University
Review, Operations Research, etc.

Disclaimer

The conclusions and opinions expressed in this
document are those of the author cultivated in the freedom of expression,
academic environment of Air University. They do not reflect the official
position of the U.S. Government, Department of Defense, the United States Air
Force or the Air University.